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Worksheet: Elastic Potential Energy

In this worksheet, we will practice using F = kx (Hooke's law) and E = ½kx² to find the force and the potential energy of a compressed spring.

Q1:

In the movie Monty Python and the Holy Grail, a cow of mass 110 kg
is catapulted from the top of a castle wall 9.1 m high, onto the people
down below. The cow is launched from a spring of with a constant
of 1.1×104 N/m
that was initially compressed by 0.50 m from equilibrium.
The gravitational potential energy is set to zero at the base of the castle wall.

What is the gravitational potential energy of the cow as it just clears the castle wall, treating the cow as a point particle?

A11×103 J

B8.9×103 J

C12×103 J

D9.8×103 J

E13×103 J

What is the elastic spring energy of the cow before the catapult is released?

A1.4×103 J

B1.2×103 J

C0.89×103 J

D0.65×103 J

E1.5×103 J

What is the speed of the cow at the instant that it hits the ground?

Q2:

A box that has 50 J of kinetic energy
slides on a frictionless surface. The box hits a spring and compresses the spring a distance
of 25 cm from its equilibrium
length. If the same box with the same initial energy slides on a rough surface, it only
compresses the spring a distance of
15 cm. How much energy must have
been lost by the box when sliding on the rough surface?

Q3:

The spring of a spring gun has a force constant 𝑘=12/Ncm.
When the gun is aimed vertically upward,
a 15-g projectile is shot to a height of 5.0 m above the end of the expanded spring,
as shown in the accompanying diagram. How much was the spring compressed initially?

Q4:

A pogo stick has a spring constant of
2.50×104 N/m.
A child stands on the pogo stick,
pointing the stick vertically upward and compressing it by 12.0 cm.
The child and pogo stick have a combined mass of 40.0 kg. The child jumps,
which exerts negligible force but causes the pogo stick to return to its equilibrium length.
What is the maximum upward vertical displacement of the child?

Q5:

A block of mass 230 g is attached to one end of a spring that
has a spring constant of 120 N/m. The other end of the spring is
attached to a support while the block rests on a smooth horizontal
table and can slide freely without any friction. The block is
pushed horizontally to compress the spring by 12 cm.
The block is then released from rest.

How much potential energy is stored in the spring when the spring is fully compressed?

Determine the speed of the block when the spring has recoiled to its equilibrium length.

Q6:

You compress a spring by 𝑥 and then
release it. Next, you compress the spring by 2𝑥. How much more work did you do
the second time than the first?

AHalf as much.

BA quarter as much.

CThe same.

DFour times as much.

ETwice as much.

Q7:

You are loading a toy dart gun, which has
two settings. The spring in the toy gun is compressed twice as far in the more
powerful setting as it is in the less powerful setting. If it takes 5.0 J
of
work to compress the dart gun to the lower setting, how much work does it take
for the higher setting?

Q8:

A bungee cord exerts a nonlinear elastic restoring force of magnitude 𝐹(𝑥)=188𝑥/−0.672𝑥/NmNm33
when it is extended. How much work is required to extend the bungee cord by 12.7 m?

Q9:

In a Coyote/Road Runner cartoon clip, a spring expands quickly and sends the coyote into a rock. The spring extends by 5.0 m and accelerates the coyote of mass 20 kg
to a speed of 15 m/s.

What is the spring constant of this spring?

If the coyote were sent vertically upward with the energy given to him by the spring, what maximum height would it reach, assuming air resistance was negligible?

Q10:

A Delorean car of mass 1230 kg
travels at 88 mph.
What spring constant would a spring need to have to bring the car to rest in a distance of
0.25 m?

A3.2×107 N/m

B1.7×107 N/m

C5.0×107 N/m

D3.0×107 N/m

E9.0×107 N/m

Q11:

A child of mass 32 kg jumps up and down on a trampoline.
The trampoline exerts a spring restoring force on the child with a
spring constant of 5.0×103 N/m.
At the highest point of the bounce,
the child is 1.0 m vertically above the unstretched surface level of the trampoline.
What distance is the trampoline compressed by when the child jumps on it?
Neglect the bending of the child’s legs or any transfer of energy of the child into the trampoline while jumping.

Q12:

A perfectly elastic spring with an equilibrium length of
20.0 cm and a spring
constant of 400.0 N/m
changes in length from
22.0 cm in such a way
that the elastic potential energy stored in the spring increases by
0.0800 J. What is the
length of the spring after its length changes?

Q13:

A perfectly elastic spring has an equilibrium length of
0.20 m and a spring constant
of 0.400 kN/m. The
spring is extended so that its length becomes
0.23 m.

How much elastic potential energy is contained in the spring when it is
extended?

The spring is further extended so that its length becomes 0.26 m. What is the increase
in the elastic potential energy due to the additional extension?

Q14:

A massless spring with force constant
𝑘=200/Nm
hangs from the ceiling.
A
2.0 kg
block is attached to the
free end of the spring and released.
If the block falls 17 cm
before starting back upwards, how much work is done by
friction during its descent?

Q15:

386 J of work is required to compress a spring by 117 mm. What is the force constant of the spring?

A4.66×103 N/m

B3.2×104 N/m

C3.04×104 N/m

D5.64×104 N/m

E7.43×104 N/m

Q16:

A spring has a force constant of 53.0 N/m.
An object, initially at rest, with a mass of 0.960 kg is suspended from it.
The object descends, stretching the string, oscillates, and then comes to rest.

How much is the spring stretched when the object
has come to rest after oscillating?

Calculate the decrease in the gravitational potential energy of the object between its position
at the point at which it is attached to
the unextended spring and its position at the point at which it comes to rest after oscillating.

Calculate the energy stored in the spring by its
extension.

Q17:

Old-fashioned pocket watches needed to be wound daily so they would not run down and lose
time due to the friction in the internal components. This required a large number of turns
of the winding key, but not much force per turn, and it was possible to overwind and break
the watch. In which of the following ways was the energy stored?